A mathematical phrase that contains a variable is an open phrase. A number phrase is an expression that does not contain a variable. It is also referred to as a numerical expression. The English phrase ‘a certain number added to 5’ is translated to an open phrase ‘n + 5’ where n stands for a certain number. The English phrase ‘seven added to 5’ is translated to a number phrase ‘7 + 5’.
Expressions like 8 + 2, 12 – 2, 5 x 2, and 20 ÷ 2 are some number phrases for the number 10. Expressions like , - 2a, 2n + 8, 2(l + w) are examples of open phrases. Another name for open phrase is algebraic expression. An expression composed of constants, variables, grouping symbols, and operation symbols, is called an algebraic expression.
In an open phrase “-7x + 12”, -7x and 12 are the terms of the expression. In the term -7x, -7 is the constant, also called the numerical coefficient of x while x is the variable, or the literal coefficient of -7. The numerical coefficient of a term is written before the literal coefficient. A term is an indicated product or quotient of coefficients. The term 12 is the constant term, which does not have any indicated literal coefficient. Terms in an algebraic expression are separated by the plus (+) or minus (-) sign.
The symbols of operations and relations used in algebra are as follows:
Symbol Meaning
+ addition, plus, increased by, added to, the sum of, more than
- subtraction, minus, decreased by, subtracted from, less than, diminished by
· , ( ) multiplication, times, multiplied by
/ , ÷ division, divided by, ratio of, the quotient of
= equals, is equal to
< is less than
> is greater than
≤ is less than or equal to
≥ is greater than or equal to
≠ is not equal to
Translate each verbal phrase into an algebraic expression.
Verbal Phrase Algebraic Expression
1. Sum of two numbers a + b
2. Twice a certain number 2n
3. Difference of 8 and thrice a certain number 8 – 3b
4. Quotient of a number and 3 diminished by 2 – 2
5. Square of the sum of a number and 2 (x + 2)2
Any letter or variable can represent the numbers because the numbers are not known. In item #1, a and b are the symbols used to represent the said numbers. We add them because of the word ‘sum’, hence the use of the plus sign (+).
Item #2, the variable n represents the number. The phrase is translated as the product of 2 and n.
Item #3, the variable b represents the number and is used as a coefficient of 3. Their product is subtracted from 8 because of the word ‘difference’.
Item #4, the variable x represents the number with 3 as its denominator because of the word ‘quotient’. The phrase ‘diminished by 2’ is translated as – 2.
Item #5, the variable represents the number added to 2. Because of the word square, the quantity (x + 2) is raised to the second power.
Translate each verbal phrase into an algebraic expression.
Verbal Phrase Algebraic Expression
A number y increased by four y + 4
Three times a number m decreased by 6 3m – 6
Nine added to the quotient of m and five 9 + (m/5)
Translate each algebraic expression into a verbal phrase.
Algebraic Expression Verbal Phrase
8n – 12 a) The difference between eight times the number n and twelve
b) The product of eight and the number n minus twelve
c) Eight times the number n decreased by twelve
d) Twelve less than eight times the number n.
2(x + y) a) Twice the sum of the numbers x and y
b) Two times the sum of the numbers x and y
c) Two multiplied by the sum of the numbers x and y
d) The product of two and the sum of two numbers x and y
x2 + 3 a) The sum of the square of the number x and three
b) The square of the number x increased by three
c) Three more than the square of the number x
d) Three added to the square of the number x
To translate a verbal sentence into a mathematical sentence, we use a symbol that corresponds to every word or phrase in the given verbal sentence until the mathematical sentence is formed
To translate a mathematical sentence into a verbal sentence, we use a word or a phrase to that corresponds to every symbol in the given mathematical sentence until the thought of the verbal sentence is obtained.